On Universes in Type Theory

Transcription

1 On Universes in Type Theory 1 Introduction Erik Palmgren 1 Uppsala University The notion of a universe of types was introduced into constructive type theory by Martin-Löf (1975) According to the propositions-as-types principle inherent in type theory, the notion plays two rôles The first is as a collection of sets or types closed under certain type constructions The second is as a set of constructively given infinitary formulas In this paper we discuss the notion of universe in type theory and suggest and study some useful extensions We assume familiarity with type theory as presented in eg (Martin-Löf 1984) Universes have been effective in expanding the realm of constructivism One example is constructive category theory where type universes take the rôles of Grothendieck universes of sets, in handling large categories A more profound example is Aczel s (1986) type-theoretic interpretation of constructive set theory (CZF) It is done by coding -diagrams into well-order types, with branching over an arbitrary type of the universe The latter generality is crucial to interpret the separation axiom The introduction of universes and well-orders (W-types) in conjunction gives a great proof-theoretic strength This has provided constructive justification of strong subsystems of second order arithmetic studied by proof-theorists (see Griffor and Rathjen (1994) and Setzer (1993), and for some early results, see Palmgren (1992)) At present, it appears that the most easily justifiable way to increase the proof-theoretic strength of type theory is to introduce ever more powerful universe constructions We will give two such extensions in this paper Besides contributing to the understanding of subsystems of second order arithmetic and pushing the limits of inductive definability, such constructions provide intuitionistic analogues of large cardinals (Rathjen et al to appear) A third new use of universes is to facilitate the incorporation of classical reasoning into constructive type theory We introduce a universe of classical propositions and prove a conservation result for Π 2 -formulas Extracting programs from classical proofs is then tractable within type theory The next section gives an introduction to the notion of universe The central part of the paper is Section 3 where we introduce a universe forming operator and a super universe closed under this operator Section 4 summarises what is known 1 The research reported herein was supported partly by the Swedish Research Councils for Natural Sciences (NFR) and Engineering Sciences (TFR), and partly by the EU Project Twinning: Proof Theory and Computation (contract SC1*-CT (TSTS)) Author s current address: Department of Mathematics, Chalmers University of Technology and the University of Göteborg, S Göteborg 1

2 2 On Universes in Type Theory about the proof-theoretic strength of this extension, mainly results due to M Rathjen In Section 5 we introduce the notion of higher order universe operators While all of the preceding development is predicative, it is also possible to define impredicative theories using universes In Section 6 we point out some dangers in combining such ideas with elimination rules In particular, we discuss Setzer s Mahlo universe Finally, in Section 7 we construct the classical universe 2 Universes From an abstract point of view a type universe is simply a type of types closed under certain type constructions Being a type of types can be formulated in essentially two ways (Martin-Löf 1984): à la Tarski, by introducing a type of codes U for types and a decoding function T( ): U type a U T(a) type or, alternatively, à la Russell by simply introducing U and identifying codes and types A U U type A type The Russell formulation should be regarded as an informal version of the Tarski formulation, but is too unclear when nesting universe constructions, eg as in the super universe Thus we use the Tarski formulation for complete precision Modern presentations of type theory employ a so called logical framework (Nordström et al 1990) This a typed lambda calculus with a dependent function space construction (Π-types) and a universe of types (Set, El( )) The types of this universe are called sets In this framework different type theories can be specified by giving closure conditions to the sets, and by introducing constants and computation rules to types constructed from sets Later extensions have also Σ-types or records We shall here present the rules for the extensions of type theory in the older, more readable style of Martin-Löf (1984), as far as possible (In section 5 we need however a logical framework with Σ-types) In Martin-Löf s type theory two different conceptions of universes occur The first captures the idea of reflection of the judgement forms A set and A = B into a hierarchy of universes (U n, T n ) externally indexed by n = 1, 2, 3, That is, whenever A set then in some universe U n, there is a code a so that T n (a) = A, and if A = B with T n (a) = A and T n (b) = B, then a = b U n This is as in Martin-Löf (1975; 1982) albeit there it is formulated à la Russell The second idea, which is preferable, is to uniformly construct universes above earlier universes (hinted at in Martin-Löf (1984), p 89) Universes as full reflections We view the formation of the hierarchy (U 1, T 1 ), (U 2, T 2 ), as a process At first there is no universe Then we introduce a universe U 1 of codes for all basic sets, U 1 set x U 1 T 1 (x) set

3 On Universes in Type Theory 3 n 1 0, n 1 1, n 1 U 1 T 1 (n 1 0) = N 0 T 1 (n 1 1) = N 1 T 1 (n 1 ) = N, where N 0 is the empty set, N 1 is the set with single element 0 1, and N is the set of natural numbers Furthermore we assume that it is closed under Π-formation: (x T 1 (a)) a U 1 b U 1 π(a, (x)b) U 1 (x T 1 (a)) a U 1 b U 1 T 1 (π(a, (x)b)) = (Πx T 1 (a))t 1 (b), and we also assume that (U 1, T 1 ) is similarly closed under Σ, I, + and other set formers, if desired Hence for every set A formed without universes there is a U 1 so that T 1 (a) = A At this stage there is no difference between the two versions of universes We have that if A = B is formed without the use of universes, then a = b U 1 for some a and b such that T 1 (a) = A, T 1 (b) = B, by the usual equalities that come with every canonical constant Then at the next stage we introduce a new universe (U 2, T 2 ) closed under the set formers Π, Σ, with new codes for all sets n 2 0, n2 1, n2 U 2 T 2 (n 2 0 ) = N 0 T 2 (n 2 1 ) = N 1 T 2 (n 2 ) = N But U 1 and T 1 (x) are sets of the previous stage, so we must also introduce u 2 1 U 2 T 2 (u 2 1 ) = U 1 x U 1 t 2 1 (x) U 2 We have a host of new set equalities to reflect in U 2 : T 1 (n 1 k ) = N k and T 1 (n 1 ) = N give t 2 1(n 1 k) = n 2 k U 2 t 2 1(n 1 ) = n 2 U 2 and since T 1 (π(a, (x)b)) = (Πx T 1 (a))t 1 (b) we should assume t 2 1(π(a, (x)b)) = π(t 2 1(a), (x)t 2 1(b)) U 2 and so on for all codes for set formers We may also express this as: t 2 1 is a homomorphism with respect to set constructors, extending t 2 1(n 1 k ) = n2 k and t2 1(n 1 ) = n 2 U 2 At each step in the construction of the hierarchy of universes we introduce new codes and equalities between codes for sets and set equalities which can be formed Having completed the hierarchy (U 1, T 1 ), (U 2, T 2 ),, (U n, T n ),, n < ω we notice that any proof in the resulting system can only use finitely many universes (proofs are finite and the universes are externally indexed) and hence the reflection principle holds for both sets and set equalities If we try to iterate this process into the transfinite we run into something like u ω 1, u ω 2, u ω 3, U ω a universe which has infinitely many introduction rules Thus it is impossible to formulate an elimination rule without having some kind of internal indexing of the universes

4 4 On Universes in Type Theory Universes as uniform constructions Here we do not reflect set equalities This allows us to simply inject the codes for sets from an earlier universe into the next We construct a hierarchy of universes (U 1, T 1 ), (U 2, T 2 ), stepwise Assume that (U n, T n ) has already been constructed, then and U n+1 set u n U n+1 x U n+1 T n+1 (x) set T n+1 (u n ) = U n x U n t n (x) U n+1 x U n T n+1 (t n (x)) = T n (x) Thus t n (a) is now considered to be a canonical element in U n+1, and is regarded as a copy of a in U n+1 As an example, note that the code for U j, j < n + 1 in U n+1 is t n (t n 1 ( t j+1 (u j ))) We furthermore assume that (U n+1, T n+1 ) is closed under the same set formers as (U n, T n ) The construction of (U n+1, T n+1 ) depends thus only on the family (U n, T n ) Observe that we still reflect the judgement form A set It seems that the idea of universes as full reflections is difficult to formulate for transfinite hierarchies The usefulness of reflecting equalities of sets is not clear Thus we shall only consider hierarchies of universes built using the uniform construction Remark The formation of the next universe was formulated as an operator in the domain-theoretic model (Palmgren 1993) of partial type theory This leads to the formalisation of universe operators in the next section 3 Universe Operators and Super Universes By having universe formation as an operator and a super universe closed under this operator we may form transfinite level universes much the same way as we may form transfinite sets using an ordinary universe The universe forming operation acts on families of sets We can form a universe (U(A, (x)b), T(A, (x)b)) above any family of sets (A, (x)b) (x A) A set B set U(A, (x)b) set a U(A, (x)b) T(A, (x)b, a) set We assume that (U(A, (x)b), T(A, (x)b)) is closed under the usual set formers Π, Σ, +, Id That the universe is above the family (A, (x)b) is expressed by (A, (x)b) U(A, (x)b) T(A, (x)b, (A, (x)b)) = A

5 On Universes in Type Theory 5 a A l(a, (x)b, a) U(A, (x)b) a A T(A, (x)b, l(a, (x)b, a)) = B(a/x) Thus (A, (x)b) is a code for A in U(A, (x)b), and l(a, (x)b, a) is a copy of the code a in A for B(a/x) If we assume that (U 0, T 0 ) is some basic family of sets, then we can define the hierarchy of universes as follows U n+1 := U(U n, (x)t n (x)) T n+1 (a) := T(U n, (x)t n (x), a), and let u n := (U n, (x)t n (x)) and t n (a) := l(u n, (x)t n (x), a) The super universe We now consider a universe (V, S) the super universe which in addition to being closed under the set formers Π, Σ, +, I is also closed under universe formation Moreover we assume that it contains basic sets The closure under the universe operator is given by (x S(a))) a V b V u(a, (x)b) V (x S(a))) a V b V S(u(a, (x)b)) = U(S(a), (x)s(b)) (x S(a)) a V b V c S(u(a, (x)b)) t(a, (x)b, c) V (x S(a)) a V b V c S(u(a, (x)b)) S(t(a, (x)b, c)) = T(S(a), (x)s(b), c) Note that u(a, (x)b) and t(a, (x)b, c) are canonical elements The term t(a, (x)b, ) injects codes from the universe U(S(a), (x)s(b)) into V The super universe has an inductive structure and it is not difficult to formulate an elimination rule for it Transfinite hierarchies Examples of a transfinite sets can easily be constructed using recursion and a universe (cf Martin-Löf (1975), p 83) Transfinite level universes are, however, more complicated to construct since they are to be given as families of sets Consider the set of all codes for families in the super universe V F V := (Σx V )[S(x) V ] In the following, let, denote pairing and p,q the first and second projection respectively Define B V (c) := S(p(c)) for c F V, the base of the family coded by c, and F V (c, x) := S(Ap(q(c), x)) for x B V (c), the family of sets over B V (c) coded by c F V We shall define û F V F V, such that B V (û(c)) = U(B V (c), (x)f V (c, x)) F V (û(c), w) = T(B V (c), (x)f V (c, x), w)

6 6 On Universes in Type Theory and this is achieved by û := (λc) u(p(c), (x)ap(q(c), x)), (λw)t(p(c), (x)ap(q(c), x), w) Hence if c is a code for a universe, then û(c) is a code for the universe above c Let c 0 be a code for a suitable basic family of sets By recursion we can define û n (c 0 ) (n N) the codes for finite iterates of universes above c 0 Then U ω := U((Σn N)B V (û n (c 0 )), (z)f V (û p(z) (c 0 ), q(z))) is a universe of transfinite level, and it is straightforward to find its code in the super universe 4 Proof-theoretic Strength Type theory with one universe closed under the W-set, named ML 1 W, is prooftheoretically very strong, among the theories that have so far been given a complete constructive justification Recall that the W-set is a general inductive set former, by which one may construct the Brouwer ordinals as well-founded trees which branch over a given family of sets A slight weakening of ML 1 W has the strength of Kripke Platek set theory extended with a principle corresponding to the existence of an inaccessible cardinal (Griffor and Rathjen 1994) Independently, Setzer (1993) determined the strength of the full theory It is interesting to note that universes give strength already without W-sets Let γ 0 = ε 0 and let γ n+1 = ϕ γn 0, where (ϕ α ) α are the Veblen functions, ie ϕ 0 (ξ) = ω ξ and, for α > 0, ϕ α is the enumeration function for the common fixed points of the functions ϕ β (β < α) Aczel (1977) showed that the strength of type theory with one universe is γ 1 Hancock s conjecture (cf Martin-Löf (1975)) stated that the strength of type theory with n universes is γ n, and was proved by Feferman (1982) From this it follows that the strength of type theory with arbitrarily many finite level universes is the limit of (γ n ) n, ie Γ 0 The latter result was achieved independently by Aczel In a previous version of the present paper we interpreted an intuitionistic version of a theory ATR using an internally indexed hierarchy (U n, T n ) (n N) The classical version of this theory has strength Γ ε0 (cf Simpson 1982) Subsequently, Rathjen has obtained sharp results for theories involving the super universe One ingredient in the proof of the lower bound of the super universe is (a relativised version of) the interpretation of ATR We summarise his results Let MLU denote the type theory with the universe operator U of Section 3 and no elimination rules for U Let MLS be type theory with the universe operator U and the super universe closed under this operator, as in Section 3, and no elimination rules The variant of MLS where the operator U may only act on families from the super universe is called MLS Let (Φ α ) α be defined just as the hierarchy of Veblen functions, except that Φ 0 (ξ) = Γ ξ Theorem 41 (Rathjen 1997)

7 On Universes in Type Theory 7 (i) MLU = Γ 0 (ii) MLS = Φ ε0 (0) (iii) MLS = Φ Γ0 (0) The strength of MLS with W-types has also been determined by Rathjen (1997) We refer to Griffor and Rathjen (1994), Palmgren (1992), Rathjen et al (to appear), Setzer (1993; 1995) and the next section for further proof-theoretic results Remark The ordinal Γ 0 is usually called the Feferman Schütte bound for predicativity The proof-theorist s notion of predicativity is based on the idea that an ordinal is predicative if it can be reached by a certain autonomous progression of theories starting from Peano arithmetic This is to be contrasted with what we could call the constructivist s notion of predicativity, which recognises a construction as predicative if it has a clear inductive structure, eg W-sets and super universes Note for example that the theory MLS goes well beyond Γ 0 Not too many theories of strength between Γ 0 and the Howard ordinal have been found According to the results above, universes seem to provide natural examples of such theories 5 Higher Order Universe Operators The notion of universe operator can be extended to all finite orders To formulate them we use a logical framework (Set, El( )) with Σ-types The Σ-types are written in boldface (Σx B)C and their associated pairing function, left and right projections are denoted by,, p and q respectively Where no confusion can arise we write A instead of El(A) to simplify the presentation Definition 51 Construct an externally indexed hierarchy of types O 0 = Set, F n = (ΣA V )(A)O n, O n+1 = (F n )F n Then O n is the type of operators of order n, and F n is the type of families of operators of order n The Theories ML n, n = 0,1,2, We define this sequence of theories inductively Basic type theory with Π, Σ, + and I-sets, and the basic sets N 0, N 1 and N is ML 0 We define a type theory ML n+1 by adding to the theory ML n, the new functions U0 n,, Un n, T 0 n,, T n n, ln 0,, ln n, n 0,, n n, un 0,, un n 1 and t n 0,, t n n 1 The pair Uk n, T k n is used to construct a family of operators of order k from given families operators of orders k, k + 1,,n Thus U0 n, T 0 n will construct the actual universe The constants l n k, n k are lifting functions analogous to l and for the universe operator of Section 3 The constants u n k, tn k signify the application of an operator of level k to a family of operators of level k 1 All these functions are canonical (constructors), except the Tk n :s Their axiomatisation is as follows

11 On Universes in Type Theory 11 We note that the universe (Ü, T) is in some sense non-wellfounded Indeed, assume one imposes the natural elimination rule for the universe by assuming U-elimination (cf Palmgren (1992), p 95) extended with a clause for the -case h( (a, (X)b)) = d (a, (X)b, h(a), (λx)h(b)) (61) Then we obtain an inconsistent theory with non normalising terms Let n 0 and n 1 be codes for the sets N 0 and N 1, respectively Using (61) we define a term h(z) Ü (z Ü) such that h( (a, (x)b)) = Ap(g, Ap((λx)h(b), (λy)ϕ)) where g Ü Ü is an arbitrary function and ϕ Ü Now letting ϕ := (n 1, (x)ap(x, 0 1 )), we have h(ϕ) = Ap(g, Ap((λx)h(Ap(x, 0 1 )), (λy)ϕ)) = Ap(g, h(ϕ)) Hence h(ϕ) is a fixed point of g Letting g := (λx)(x n 0 ), this leads to outright inconsistency, since we then obtain a set A = T(h(ϕ)), such that A = A If we instead take g := (λx)(x x), the equation A = A A emerges From this we obtain a nonterminating term, by considering it as a model of untyped λ-calculus The problematic point with the above universe is that it occurs negatively in one of its own introduction rules Another, proof-theoretically more interesting example, is Setzer s Mahlo universe (M, S) (Setzer 1995) Here one crucial introduction rule is f F M F M u f M, where F M = (Σx M)[S(x) M] Similarly to the above we can prove that it is inconsistent with the natural elimination rule This rule is analogous to the one for (Ü, T) but we have instead h(u f ) = d u (f, (λx)h(f(x))) For any g M M we define g + F M F M by g + (w) = g(p(w)), (λx)n 1, and for any f F M F M we define f M M by f (a) = p(f a, (λx)n 1 ) (The particular choice of n 1 is not important, any other code would do) Thus (g + ) (a) = g(a) By the natural elimination rule, there exists h M M such that h(u f ) = h(f (u f )) n 0 (f F M F M ) Now put f = ((λx M)x) + Then by the above f (u f ) = u f, so h(u f ) = h(u f ) n 0 Hence A = A, for some A and analogously to the above B = B B for some B We summarise the results as a theorem Theorem 61 Let T be a type theory with either the second order universe or with Setzer s Mahlo universe Then T becomes inconsistent and non-normalising, when adding the natural elimination rules We remark that Setzer did not himself consider an elimination rule for his universe (Setzer 1995) However, it seems reasonable from a predicative point of view to require that any set introduced in type theory should be consistent with the natural elimination rules generated by the introduction rules

12 12 On Universes in Type Theory 7 Classical Universes Within Type Theory The A-translation is a combination of Gödel s negative translation with Dragalin and Friedman s wellknown syntactic translation This translation gives an easy method for proving conservativity of Π 0 2-sentences of many classical theories over their intuitionistic counterpart We shall here use a universe of classical propositions to obtain a semantic version of this method The idea is to extend type theory with a universe of propositions for which classical logic holds It is in a precise sense a (small-) complete boolean algebra with prescribed falsity The A-translation for the,, fragment of minimal logic (Berger and Schwichtenberg 1995) has a particularly simple form We shall make a semantic version of this translation Our starting point is a Martin-Löf type theory with a universe of sets (U, T) Here it will be useful to think of a code a U as a (constructively) given infinitary formula, and the decoding T(a) as its canonical Tarski semantics We extend this type theory with a universe of propositions (U, T A ) for each set A We define it as follows: U set A set b U T A (b) set The absurdity of this universe will be A, and for each set p of U we introduce a new proposition g p into the new universe U T A ( ) = A p U g p U p U T A ( g p) = (T(p) A) A We assume closure only under implication, conjunction and universal quantification over small sets: b U c U b c U b U c U T A (b c) = T A (b) T A (c) b U c U b c U b U c U T A (b c) = T A (b) T A (c) (x T(s)) s U b U (s, (x)b) U (x T(s)) s U b U T A ( (s, (x)b)) = (Πx T(s))T A (b) We admit also proof by induction on this universe, a principle which is no stronger than recursion on an ordinary universe This is then the semantic version of the A-translation for minimal logic The basic results are proved similarly as in the syntactic case

13 On Universes in Type Theory 13 Theorem 71 The universe (U, T A ) satisfies stability and ex falso quod libet, ie there are constructions for T A ( b b) and T A ( b) for any b U Proof By induction on the universe Theorem 72 (Π 2 -conservation) Let p(x, y) U (x R, y S) be a family of small sets over small sets R = T(r) and S = T(s) If for all small A, T A ( (r, (x) (s, (y) g p(x, y)))) is true, then (Πx R)(Σy S)T(p(x, y)) is true Proof For any given x R, substitute for A the set (Σy S)T(p(x, y)) and then proceed as in the familiar syntactic proof Note that T A (b) does not in general follow from T(b) It is possible to make intricate analyses for what b this in fact is the case, by generalising results from the syntactic situation Here we only observe that T A ( g i(s, x, y)) holds whenever I(T(s), x, y) holds and that the translation of Peano s fourth axiom (n + 1 0) is valid Moreover the induction schemata for natural numbers and W-sets, with branching over any small family of sets, are valid in translated form This means that in the classical universe we may use higher type arithmetic and the mentioned induction schemes It seems to be an interesting task to investigate what further principles are valid The semantic version of the A-translation was completely formalised using the proof support system ALF, and tested on a small program extraction problem This was done in cooperation with U Berger The advantage of the semantic version is that it is possible to work entirely within one theory, and that classical and constructive methods may be mixed 8 Acknowledgements I am grateful to Per Martin-Löf for encouragement to pursue the construction of universes, and to Ed Griffor and Michael Rathjen for discussions Section 2 and 3 formed part of my PhD Thesis Section 7 was written (and implemented) while visiting the Ludwig-Maximilians Universität in Munich Thanks go to Helmut Schwichtenberg, Ulrich Berger and Anton Setzer for their hospitality I thank Peter Hancock and an anonymous referee for valuable comments on the content and presentation of this paper Bibliography Aczel, P (1977) The strength of Martin-Löf s intuitionistic type theory with one universe In: S Miettinen and J Väänänen (eds) Proc Symp on Mathematical Logic (Oulo 1974), pp 1 32, Report no 2, Dept of Philosophy, University of Helsinki Aczel, P (1986) The type theoretic interpretation of constructive set theory: inductive definitions In: RB Marcus et al (eds) Logic, Methodology and Philosophy of Science VII, pp North-Holland, Amsterdam Berger, U and Schwichtenberg, H (1995) Program extraction from classical proofs In: D Leivant (ed) Logic and Computational Complexity, Indianapolis

CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes

A Propositional Dynamic Logic for CCS Programs Mario R. F. Benevides and L. Menasché Schechter {mario,luis}@cos.ufrj.br Abstract This work presents a Propositional Dynamic Logic in which the programs are

1 Fixed-Point Logics and Computation Symposium on the Unusual Effectiveness of Logic in Computer Science University of Cambridge 2 Mathematical Logic Mathematical logic seeks to formalise the process of

Intuitionistic Type Theory Per Martin-Löf Notes by Giovanni Sambin of a series of lectures given in Padua, June 1980 Contents Introductory remarks............................ 1 Propositions and judgements.......................

Degrees that are not degrees of categoricity Bernard A. Anderson Department of Mathematics and Physical Sciences Gordon State College banderson@gordonstate.edu www.gordonstate.edu/faculty/banderson Barbara

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 2012 1 p. 43 48 ON FUNCTIONAL SYMBOL-FREE LOGIC PROGRAMS I nf or m at i cs L. A. HAYKAZYAN * Chair of Programming and Information

THE GENERALISED TYPE-THEORETIC INTERPRETATION OF CONSTRUCTIVE SET THEORY NICOLA GAMBINO AND PETER ACZEL Abstract. We present a generalisation of the type-theoretic interpretation of constructive set theory

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

From second order Analysis to subsystems of set theory Dedicated to Gerhard Jäger on the occasion of his 60th birthday Wolfram Pohlers December 13, 2013 1 Introduction It is a real pleasure for me to be

THE ω-enumeration DEGREES IVAN N. SOSKOV Abstract. In the present paper we initiate the study of the partial ordering of the ω-enumeration degrees. This ordering is a semi-lattice which extends the semi-lattice

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

Dependent Types at Work Ana Bove and Peter Dybjer Chalmers University of Technology, Göteborg, Sweden {bove,peterd}@chalmers.se Abstract. In these lecture notes we give an introduction to functional programming

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

12. Large cardinals The study, or use, of large cardinals is one of the most active areas of research in set theory currently. There are many provably different kinds of large cardinals whose descriptions

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

EMBEDDING COUNTABLE PARTIAL ORDERINGS IN THE ENUMERATION DEGREES AND THE ω-enumeration DEGREES MARIYA I. SOSKOVA AND IVAN N. SOSKOV 1. Introduction One of the most basic measures of the complexity of a

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

Chapter 1 Verificationism Then and Now Per Martin-Löf The term veri fi cationism is used in two different ways: the fi rst is in relation to the veri fi cation principle of meaning, which we usually and

Predicate Logic Review UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Grammar A term is an individual constant or a variable. An individual constant is a lowercase letter from the beginning

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

Gödel s correspondence on proof theory and constructive mathematics W. W. Tait The volumes of Gödel s collected papers under review consist almost entirely of a rich selection of his philosophical/scientific

ON THE EMBEDDING OF BIRKHOFF-WITT RINGS IN QUOTIENT FIELDS DOV TAMARI 1. Introduction and general idea. In this note a natural problem arising in connection with so-called Birkhoff-Witt algebras of Lie

Regular Languages and Finite Automata 1 Introduction Hing Leung Department of Computer Science New Mexico State University Sep 16, 2010 In 1943, McCulloch and Pitts [4] published a pioneering work on a

ASSOCIATIVE-COMMUTATIVE REWRITING* Nachum Dershowitz University of Illinois Urbana, IL 61801 N. Alan Josephson University of Illinois Urbana, IL 01801 Jieh Hsiang State University of New York Stony brook,

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

Predicate Logic Example: All men are mortal. Socrates is a man. Socrates is mortal. Note: We need logic laws that work for statements involving quantities like some and all. In English, the predicate is

A Natural Deduction System Preserving Falsity 1 Wagner de Campos Sanz Dept. of Philosophy/UFG/Brazil sanz@fchf.ufg.br Abstract This paper presents a natural deduction system preserving falsity. This new

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

Truth as Modality Amsterdam Workshop on Truth 13 March 2013 truth be told again Theodora Achourioti ILLC University of Amsterdam Philosophical preliminaries i The philosophical idea: There is more to truth

Low upper bound of ideals, coding into rich Π 0 1 classes Antonín Kučera the main part is a joint project with T. Slaman Charles University, Prague September 2007, Chicago The main result There is a low

ICMS, 26 May 2007 1/17 Bindings, mobility of bindings, and the -quantifier Dale Miller, INRIA-Saclay and LIX, École Polytechnique This talk is based on papers with Tiu in LICS2003 & ACM ToCL, and experience

A SURVEY OF RESULTS ON THE D.C.E. AND n-c.e. DEGREES STEFFEN LEMPP 1. Early history This paper gives a brief survey of work on the d.c.e. and n-c.e. degrees done over the past fifty years, with particular

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

Oberwolfach, March 2 (with some later corrections) The Axiom of Univalence, a type-theoretic view point In type theory, we reduce proof-checking to type-checking Hence we want type-checking to be decidable

73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization

PRIME FACTORS OF CONSECUTIVE INTEGERS MARK BAUER AND MICHAEL A. BENNETT Abstract. This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in

Is Sometime Ever Better Than Alway? DAVID GRIES Cornell University The "intermittent assertion" method for proving programs correct is explained and compared with the conventional method. Simple conventional

A simple type system with two identity types Vladimir Voevodsky Started February 23, 201 Work in progress. 1 Introduction We call this system and its further extensions HTS for homotopy type system. It

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

Non-Archimedean Probability and Conditional Probability; ManyVal2013 Prague 2013 F.Montagna, University of Siena 1. De Finetti s approach to probability. De Finetti s definition of probability is in terms

Semantics for the Predicate Calculus: Part I (Version 0.3, revised 6:15pm, April 14, 2005. Please report typos to hhalvors@princeton.edu.) The study of formal logic is based on the fact that the validity

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negative